Algebra Examples

$\frac{9 - x^{2}}{x^{2} + 6 x + 9} \cdot \frac{3 x + 9}{3 x - 9}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $9$ as $3^{2}$ .

$\frac{3^{2} - x^{2}}{x^{2} + 6 x + 9} \cdot \frac{3 x + 9}{3 x - 9}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$\frac{(3 + x) (3 - x)}{x^{2} + 6 x + 9} \cdot \frac{3 x + 9}{3 x - 9}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$\frac{(3 + x) (3 - x)}{x^{2} + 6 x + 3^{2}} \cdot \frac{3 x + 9}{3 x - 9}$

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 2.3

Rewrite the polynomial.

$\frac{(3 + x) (3 - x)}{x^{2} + 2 \cdot x \cdot 3 + 3^{2}} \cdot \frac{3 x + 9}{3 x - 9}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{(3 + x) (3 - x)}{{(x + 3)}^{2}} \cdot \frac{3 x + 9}{3 x - 9}$

Step 3

Simplify terms.

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Step 3.1

Factor $3$ out of $3 x + 9$ .

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Step 3.1.1

Factor $3$ out of $3 x$ .

$\frac{(3 + x) (3 - x)}{{(x + 3)}^{2}} \cdot \frac{3 (x) + 9}{3 x - 9}$

Step 3.1.2

Factor $3$ out of $9$ .

$\frac{(3 + x) (3 - x)}{{(x + 3)}^{2}} \cdot \frac{3 x + 3 \cdot 3}{3 x - 9}$

Step 3.1.3

Factor $3$ out of $3 x + 3 \cdot 3$ .

$\frac{(3 + x) (3 - x)}{{(x + 3)}^{2}} \cdot \frac{3 (x + 3)}{3 x - 9}$

Step 3.2

Factor $3$ out of $3 x - 9$ .

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Step 3.2.1

Factor $3$ out of $3 x$ .

$\frac{(3 + x) (3 - x)}{{(x + 3)}^{2}} \cdot \frac{3 (x + 3)}{3 (x) - 9}$

Step 3.2.2

Factor $3$ out of $- 9$ .

$\frac{(3 + x) (3 - x)}{{(x + 3)}^{2}} \cdot \frac{3 (x + 3)}{3 x + 3 \cdot - 3}$

Step 3.2.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

$\frac{(3 + x) (3 - x)}{{(x + 3)}^{2}} \cdot \frac{3 (x + 3)}{3 (x - 3)}$

Step 3.3

Combine.

$\frac{(3 + x) (3 - x) (3 (x + 3))}{{(x + 3)}^{2} (3 (x - 3))}$

Step 3.4

Cancel the common factor of $3$ .

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Step 3.4.1

Cancel the common factor.

$\frac{(3 + x) (3 - x) (3 (x + 3))}{{(x + 3)}^{2} (3 (x - 3))}$

Step 3.4.2

Rewrite the expression.

$\frac{(3 + x) (3 - x) (x + 3)}{{(x + 3)}^{2} (x - 3)}$

Step 3.5

Cancel the common factor of $x + 3$ and ${(x + 3)}^{2}$ .

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Step 3.5.1

Factor $x + 3$ out of $(3 + x) (3 - x) (x + 3)$ .

$\frac{(x + 3) ((3 + x) (3 - x))}{{(x + 3)}^{2} (x - 3)}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $x + 3$ out of ${(x + 3)}^{2} (x - 3)$ .

$\frac{(x + 3) ((3 + x) (3 - x))}{(x + 3) ((x + 3) (x - 3))}$

Step 3.5.2.2

Cancel the common factor.

$\frac{(x + 3) ((3 + x) (3 - x))}{(x + 3) ((x + 3) (x - 3))}$

Step 3.5.2.3

Rewrite the expression.

$\frac{(3 + x) (3 - x)}{(x + 3) (x - 3)}$

Step 3.6

Cancel the common factor of $3 + x$ and $x + 3$ .

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Step 3.6.1

Reorder terms.

$\frac{(x + 3) (3 - x)}{(x + 3) (x - 3)}$

Step 3.6.2

Cancel the common factor.

$\frac{(x + 3) (3 - x)}{(x + 3) (x - 3)}$

Step 3.6.3

Rewrite the expression.

$\frac{3 - x}{x - 3}$

Step 3.7

Cancel the common factor of $3 - x$ and $x - 3$ .

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Step 3.7.1

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- 1 (- 3) - x}{x - 3}$

Step 3.7.2

Factor $- 1$ out of $- x$ .

$\frac{- 1 (- 3) - (x)}{x - 3}$

Step 3.7.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

$\frac{- 1 (- 3 + x)}{x - 3}$

Step 3.7.4

Reorder terms.

$\frac{- 1 (x - 3)}{x - 3}$

Step 3.7.5

Cancel the common factor.

$\frac{- 1 (x - 3)}{x - 3}$

Step 3.7.6

Divide $- 1$ by $1$ .

$- 1$